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An asymptotic theory for sliced inverse regression. (English) Zbl 0954.62531
Summary: Sliced Inverse Regression (S.I.R.) is a method for reducing the dimension of the explanatory variable $x$ in nonparametric regression problems. {\it K. C. Li} [J. Am. Statist. Assoc. 86, 316-342 (1991)] considers a general regression model of the form $$y=g(x'\beta_1,\dots, x'\beta_K,\varepsilon)$$ with an arbitrary and unknown link function $g$, and studies a link-free and distribution-free method for estimating $E$, the space spanned by the $\beta_k$’s, called the effective dimension reduction (e.d.r.) space. It is widely applicable, easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. The method begins with a partition of the range of $y$ into a fixed number of slices. Let us denote $T(.)$ this partition. The conditional mean of $x$ given $T(y)$ is then estimated by the sample mean within each slice.

62G05Nonparametric estimation
62J02General nonlinear regression
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